A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

Size: px
Start display at page:

Download "A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C."

Transcription

1 A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c How to find it: Try nd find limits by trditionl methods (plugging in). If you get 0 0 or!!, pply L Hopitl s rule, which sys tht lim x! c ( ) ( ) = lim x! c f x g x ( ) ( ) f " x g " x x! c. L Hopitl s rule cn be pplied whenever plugging in cretes n indeterminte form: 0 0,!!,0"!,! #!,!,0 0, nd! 0. A limit involving 0! " or " # " is found by creting quotient out of tht expression. A limit involving exponents (!,0 0, or! 0 ) involves tking nturl log of the expression to move the exponent down. e x + cos x " x ". Find lim x! 0 x 4 " x 3 A.! 30 B.! 4 C.! 6 D. 0 E. nonexistent. Find lim ( x ")ln( x ") x! + A. 0 B.! C. D. - E. nonexistent Demystifying the BC Clculus MC Exm

2 3. Find lim x! x e t " # dt x 3 " 4x A. 0 B. e4 C. e3 8 D.!e 6 E. nonexistent 4. A prticle moves in the xy-plne so tht the position of the prticle t ny time t is given by x( t) = cost nd y( t) = sin 4t. Find lim dx. t! 0 dy A. -3 B. -6 C. -8 D. -4 E. 0 t. Find the! 5. A popultion of bcteri is growing nd t ny time t, the popultion is given by " t mximum limit of the popultion. A. 500 B e C. 500e D. + ln500 E. 500e # $ Demystifying the BC Clculus MC Exm

3 B. Integrtion by Prts Wht you re finding: When you ttempt to integrte n expression, you try ll the rules you hve been given to tht point - typiclly power, substitution, nd the like. But if these don t work, integrtion by prts my do the trick. Integrtion by prts is usully used when you re need to find the integrl of product. How to find it: Integrtion by prts sttes tht " u dv = uv! " v du + C. To perform integrtion by prts, set up: u = v =. You need to fill in the u nd the dv from the originl problem. Determine du = dv = du nd v, then substitute into the formul. You re replcing one integrtion problem with nother tht might more esily be done with simple methods. The trick is to determine the u nd the dv. Functions tht cn be powered down re typiclly the u nd functions tht hve repetitive derivtives (exponentil nd trig) re typiclly the dv. 6.! x cos4x dx = A. x sin4x + cos4x + C B. x sin4x + 8 cos4x + C C. 8x sin4x + 3cos4x + C D. 8x sin4x! 3cos4x + C E. x sin4x! cos4x + C 7.! x e x dx = A. e x x! 8x +6 ( ) + C B. e x ( x! 8x + ) + C " x C. e x! x! % " x $ ' + C D. e x # &! x + %! x 3 $ $ ' + C E. e x # & + C # 4 & " 3 % Demystifying the BC Clculus MC Exm

4 8. Let R be the region bounded by the grph of y = x ln x, the x-xis nd the line x = e, s shown by the figure to the right. Find the re of R. A. e + C. e B. e! D. e E. e + 9. The shded region between the grph of y = tn! x nd the x-xis for 0 x s shown in the figure is the bse of solid whose cross-sections perpendiculr to the x-xis re squres. Find the volume of the solid. A.! + ln " B.! + e " ln 4 C.! " ln D.! 4 " ln E.! " ln 0. The function f is twice-differentible nd its derivtives re continuous. The tble below gives the vlue of f, f! nd f!! for x = 0 nd x =. Find the vlue of " x f!! ( x ) dx. x ( ) f!( x) f!! ( x) f x "5 " 4 A. -0 B. -8 C. 6 D. 4 E Demystifying the BC Clculus MC Exm

5 B. Integrtion using Prtil Frctions Wht you re finding: When you ttempt to integrte frction, typiclly you let u be the expression in the denomintor nd hope tht du will be in the numertor. When this doesn t hppen, the technique of prtil dx frctions my work. One form of this type of problem is! where x + mx + n fctors into two x + mx + n non-repeting binomils. dx How to find it: Use the Heviside method. Fctor your denomintor to get! ( x + ) ( x + b). You need to write ( x + ) ( x + b) s x + +. To find the numertor of the x + expression, cover up the x + in x + b expression, nd plug in x =!. To find the numertor of the x + b expression, cover up the ( x + ) ( x + b) x + b in expression, nd plug in x =!b. From there, ech expression cn be integrted. x + ( )( x + b).! 4x + x + 4x + 3 dx = A. ln x + 4 x C B. ln x + 4x C C. 5ln x + 3! ln x + + C D. ln x +! 5ln x C E. ln x C x +. Use the substitution u = cos x to find " sin x cos x cos x! dx. ( ) cos x! A. lncos x! + C B.!lncos x! + C C.!ln + C cos x! cos x D. ln + C E. ln cos x cos x cos x! + C Demystifying the BC Clculus MC Exm

6 3. x " 3 x! dx = A. x x + ln x + + ln x x!! + C B. + C C. x! ln x! + C D. x + ( ln x! + ln x +) + C E. ln x C x + 4. Region R is defined s the region between the grph of 9 y =, x = nd the x-xis s shown in the x + x! figure to the right. Find the re of region R. A. ln B. + 3ln4 C. 3ln 4 D. 6ln E. infinite Demystifying the BC Clculus MC Exm

7 C. Improper Integrls Wht you re finding: An improper integrl is in the form be in the form continuous. b! " f ( x ) dx or " f ( x) dx or " f ( x) dx. It lso cn! f ( x ) dx where there is t lest one vlue c such tht c b for which f x How to find it: Improper integrls re just limit problems in disguise: b # f ( x ) dx = lim!" into two pieces: b $!" b with re nd volume problems.! #!! #! ( ) is not " f ( x ) dx = lim f ( x) b#! " dx or # f ( x ) dx. In the cse where there is discontinuity t x = c, the improper integrl is split! f ( x ) dx = lim! f ( x) dx + lim! f ( x) dx. Improper integrls usully go hnd-in-hnd k" c # k" c + 5. Which of the following re convergent? k b k b I.! " dx II. x! dx III. x 0! " dx x 3 A. I only B. II only C. III only D. II nd III only E. I, II nd III " 6. # xe!4x dx = 0 A.! 6 B. 6 C. -6 D. 6 E. infinite Demystifying the BC Clculus MC Exm

8 7. The region bounded by the grph of y = 4, the line x = 4 nd the x - xis is rotted bout the x-xis. x Find the volume of the solid. A. π B. π C. 4π D. 6π E. infinite 8.! " x ( x +) dx = A.! 4 B.! C. π D. π E. infinite 4 9. To the right is grph of f ( x) =. Find the vlue of 3 " f ( x ) x! dx. ( )! A. 0 B. 3 C.! 3 D. 6 E. Divergent Demystifying the BC Clculus MC Exm

9 D. Euler s Method Wht you re finding: Euler s Method provides numericl procedure to pproximte the solution of differentil eqution with given initil vlue. How to find it: ) Strt with given initil point (x, y) on the grph of the function nd given!x = dx. ) Clculte the slope using the DEQ t the point. 3) Clculte the vlue of dy using the fct tht dy! dy dx "x. 4) Find the new vlues of y nd x: y new = y old + dy nd x new = x old +!x 5) Repet the process t step ). There re clcultor progrms vilble to perform Euler s Method. Typiclly, Euler Method problems occur in the non-clcultor section where only one or two steps of the method need to be performed. 0. Let y = f ( x) be the solution to the differentil eqution dy f 3 size of 0.5? dx = x + y with the initil condition tht ( ) =!. Wht is the pproximtion for f ( 4) if Euler s Method is used, strting t x = 3 with step A..5 B. 3.5 C. 4.5 D. 5.5 E..5. Let y = f ( x) be solution to the differentil eqution dy dx = y x with initil condition f 0 k constnt, k! 0. If Euler s method with 3 steps of equl size strting t x = 0 gives the pproximtion f 3 ( )! 0, find the vlue of k. ( ) = k, A.! B. C. D. - E.! Demystifying the BC Clculus MC Exm

10 . Consider the differentil eqution dy dx = y x the exct vlue of f 8 with initil condition f ( ) =!4. Find the difference between ( ) nd n Euler pproximtion of f ( 8) using step of 0.5. A. 0 B. C. D. 5 E (Clc) Consider the differentil eqution dy dx "!% "!% between the exct vlue of f $ ' nd n Euler pproximtion of f $ ' using two equl steps. # & # & = cos x with initil condition f ( 0) = 0. Find the difference A. 0 B C D E Demystifying the BC Clculus MC Exm

11 E. Logistic Curves Wht you re finding: Logistic curves occur when quntity is growing t rte proportionl to itself nd the room vilble for growth. This room vilble is clled the crrying cpcity. This constntly incresing curve hs distinctive S-shpe where the initil stge of growth is exponentil, then slows, nd eventully the growth essentilly stops. How to find it: Logistic growth is signled by the differentil eqution dp dt = kp ( P! t ). While this DEQ C cn be solved into P( t) =, students re not responsible for tht eqution. They need to know how to!ckt + de determine the time when the logistic growth is the fstest. This is ccomplished by d P = 0. Also students dt need to know tht the curve hs horizontl symptote mening limp t t!" ( ) = C ( the crrying cpcity). 4. A popultion of students hving contrcted the flu in school yer is modeled by function P tht stisfies the logistic differentil eqution with dp dt = P " 600! P % $ '. If P( 0) =00, find lim # 800& P( t). t!" A. 400 B. 800 C.,600 D.,400 E. 4, A popultion is modeled by function G tht stisfies the logistic differentil eqution dg dt = G " e! G % $ '. If G 0 # 4e& A. 4 B. e C. e D. 4e E. 4e ( ) =, for wht vlue of G is the popultion growing the fstest? 6. Consider the differentil eqution dy dx = ky ( L! y ). Let y = f ( x) be the prticulr solution to the differentil eqution with f ( 0) =. If x! 0, find the rnge of f ( x). A. 0,L ( ) B. ( 0,) C. ( L,] D. [,L) E. [,kl) Demystifying the BC Clculus MC Exm

12 F. Arc Length Wht you re finding: Given function on n intervl [, b], the rc length is defined s the totl length of the function from x = to x = b. For this section, we will only concentrte on curves tht re defined in function form. Functions defined prmetriclly, in polr or in vector-vlued forms hve their own formuls. How to find it: The rc length of continuous function f x b [ ] ( ) over n intervl [, b] is given by L = " + f! ( x ) dx. Most problems involving rc length need clcultors becuse of the difficulty of integrting the expression. 7. (Clc) An nt wlks round the first qudrnt region R bounded by the y-xis, the line y = x nd the curve f ( x) = 6! 4x 3 s shown in the figure to the right. Find the distnce the nt wlked. A B. 0.3 C..485 D..88 E If the length of curve from x = to x = 8 is given by! + 8x 4 dx, nd the curve psses through the point (-, 4), which of the following could be the eqution for the curve? 8 A. y =3! 9x B. y = 4! 3x 3 C. y = 7 + 3x 3 D. y =!! 3x 3 E. y = 9x! Demystifying the BC Clculus MC Exm

13 9. (Clc) The yellow bird in the populr gme Angry Birds flies long the pth y = 4 + 3x! x when x 0. When x = 4 (the point on the figure to the right), the plyer touches the screen nd the bird leves the pth nd trvels long the line tngent to the pth t tht point. If the bird crshes into the x-xis, find the totl distnce the bird flies. A B..34 C..000 D E (Clc) The grphs of i) y = x, ii) y = x nd iii) y = x! ll pss through the points (0,0) nd (,). Find the difference in rc length from the lrgest rc length to the shortest rc length of these functions on the intervl [0,]. A B. 008 C D E Find the rc length of the grph of x = ( 3 y + ) 3 for 0! y! 3. A. B. 0 C. 6 D. 3 3 E Demystifying the BC Clculus MC Exm

14 G. Prmetric Equtions Wht you re finding: Prmetric equtions re continuous functions of t in the form x = f ( t) nd y = g( t). Tken together, the prmetric equtions crete grph where the points x nd y re independent of ech other nd both dependent on the prmeter t (which is usully time). Prmetric curves when grphed do not hve to be functions. Typiclly, it is necessry to tke derivtives of prmetrics. Since the study of vectors prllels the study of prmetrics, in this section we will only nlyze the very few problems tht re not ssocited with motion in the plne. How to find it: If smooth curve C is given by the prmetric equtions x = slope of C t the point (x, y) is given by dy dy dx = dt,dx dx dt! 0. dt f ( t) nd y = g( t), then the d! dy $ The nd derivtive of the curve is given by d y dx = d # &! dy $ dt " dx % # & =. dx " dx % dx dt t = b! dx# The rc length is given by L = + dy! # % dt. The curve must be smooth nd my not intersect itself. " dt $ " dt $ t = 3. Wht is the re under the curve described by the prmetric equtions x = cost nd y = 3sin t for 0! t! "? A. 4 B. 8 C. 4 D. E A position of prticle moving in the xy-plne is given by x = t 3! 6t + 9t + nd y = t 3! 9t!. For wht vlues of t is the prticle t rest? A. 0 only B. only C. 3 only D. 0 nd only E. 0, nd Demystifying the BC Clculus MC Exm

15 34. A curve C is defined by the prmetric equtions x = t! t! 4 nd y = t 3! 7t!. Which of the following is the eqution of the line tngent to the grph of C t the point (, 4)? A. y = 6! x B. x! 4y +4 = 0 C. 5x! 3y + = 0 D. y = 4x! 4 E. No tngent line t (, 4) 35. Describe the behvior of curve C defined by the prmetric equtions x = + t nd y = t 3 + t! t! t t =. A. Incresing, concve up B. Decresing, concve up C. Incresing, concve down D. Decresing, concve down E. Incresing, no concvity 36. Find the expression which represents the length L of the pth described by the prmetric equtions x = sin ( t) nd y = cos( 3t ) for 0! t! ". " A. L = # sint cost! 3sin3t dt B. L = " 4sin 4t + 9sin 9t dt 0! C. L = " 6sin 4t cos 4t + 9sin 9t dt D. L = " 4 sin t cos t + 9sin 3t dt 0! " E. L = 6sin t cos t + 9sin 3t dt 0! 0! Demystifying the BC Clculus MC Exm

16 H. Vector-Vlued Functions Wht you re finding: While concepts like unit vectors, dot products, nd ngles between vectors re importnt for multivrible clculus, vectors in BC clculus re little more thn prmetric equtions in disguise. How to find it: Typiclly, you will be given sitution where n object is moving in the plne. You could be given either its position vector x( t) nd y( t), its velocity vector x!( t) nd y!( t) or its ccelertion vector x!! ( t) nd y!! ( t) nd use the bsic derivtive or integrl reltionships tht hve been tught in AB clculus to find the other vectors. The one formul tht students should know is tht the speed of the object is defined s the bsolute vlue of the velocity: v t ( ) = x!( t) [ ] + [ y!( t) ]. The speed is sclr, not vector. 37. A prticle moves on plne curve such tht t ny time t > 0, its x-coordinte is t! t + t 3 while its y-coordinte is! t ( ). Find the mgnitude of the prticle s ccelertion t t =. A. 4 B. C. D. 3 E The position of n object moving in the xy-plne with position function r( t) = + sint,t + cost, t 0. Wht is the mximum speed ttined by the object? A. B. C. D. 4 E. 39. A xy-plne hs both its x nd y-coordintes mesured in inches. An nt is wlking long this plne with its position vector s t 3,3t!, t mesured in minutes. Wht is the verge speed of the nt mesured in inches per minute from t = 0 to t = 3 minutes? A. B. 4 3 C. 3! D. 3 E Demystifying the BC Clculus MC Exm

17 40. An object moving in the xy-plne hs position function r( t) = the motion of the object. ( t +),t! 6ln t + ( ), t 0. Describe A. Left nd up B. Left nd down C. Right nd up D. Right nd down E. Depend on the vlue of t 4. An object moving long curve in the xy-plne hs position x( t),y t dx dy = 8t + nd = sint for t! 0. dt dt ( ( )) t time t with At time t = 0, the object is t position (5, π). Where is the object t t =!? ( ) B. (! +! + 5,! +) C.! + 5,! + A.! +! + 5,! " D.!, % $ ' E.! + 5,! # & ( ) ( ) ( ) t time t with 4. (Clc) An object moving long curve in the xy-plne hs position x( t),y ( t) dx dt = t dy + 3t + nd dt = et! for t " 0. At time t = 0, the object is t position (-6, -7). Find the position of the object t t =. A. (4.667, 3.053) B. (-3.683,.78) C. (.37, 9.78) D. (4.47, 6.053) E. (-.573, ) Demystifying the BC Clculus MC Exm

A. Limits - L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. -1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1.

A. Limits - L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. -1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1. A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by

More information

( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

More information

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

More information

AB Calculus Review Sheet

AB Calculus Review Sheet AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

More information

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.

( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists. AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find

More information

AP Calculus Multiple Choice: BC Edition Solutions

AP Calculus Multiple Choice: BC Edition Solutions AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this

More information

Topics Covered AP Calculus AB

Topics Covered AP Calculus AB Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

More information

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

More information

Math 100 Review Sheet

Math 100 Review Sheet Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

More information

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

More information

MATH 144: Business Calculus Final Review

MATH 144: Business Calculus Final Review MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

More information

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

More information

Math 31S. Rumbos Fall Solutions to Assignment #16

Math 31S. Rumbos Fall Solutions to Assignment #16 Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)

More information

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx... Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

More information

Sample Problems for the Final of Math 121, Fall, 2005

Sample Problems for the Final of Math 121, Fall, 2005 Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.

Disclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know. Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must

More information

We divide the interval [a, b] into subintervals of equal length x = b a n

We divide the interval [a, b] into subintervals of equal length x = b a n Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

More information

4.4 Areas, Integrals and Antiderivatives

4.4 Areas, Integrals and Antiderivatives . res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

More information

Review of basic calculus

Review of basic calculus Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

More information

f(a+h) f(a) x a h 0. This is the rate at which

f(a+h) f(a) x a h 0. This is the rate at which M408S Concept Inventory smple nswers These questions re open-ended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnk-out-n-nswer problems! (There re plenty of those in the

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx

CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twice-differentile function of x, then t

More information

Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivatives/Indefinite Integrals of Basic Functions Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second

More information

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral

f(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one

More information

Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED

Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type

More information

Improper Integrals, and Differential Equations

Improper Integrals, and Differential Equations Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

More information

Practice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator.

Practice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator. Nme: MATH 2250 Clculus Eric Perkerson Dte: December 11, 2015 Prctice Finl Show ll of your work, lbel your nswers clerly, nd do not use clcultor. Problem 1 Compute the following limits, showing pproprite

More information

0.1 Chapters 1: Limits and continuity

0.1 Chapters 1: Limits and continuity 1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97

More information

Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Partial Derivatives. Limits. For a single variable function f (x), the limit lim Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

lim f(x) does not exist, such that reducing a common factor between p(x) and q(x) results in the agreeable function k(x), then

lim f(x) does not exist, such that reducing a common factor between p(x) and q(x) results in the agreeable function k(x), then AP Clculus AB/BC Formul nd Concept Chet Sheet Limit of Continuous Function If f(x) is continuous function for ll rel numers, then lim f(x) = f(c) Limits of Rtionl Functions A. If f(x) is rtionl function

More information

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) = Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?

More information

How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?

How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function? Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those

More information

Overview of Calculus I

Overview of Calculus I Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,

More information

Main topics for the Second Midterm

Main topics for the Second Midterm Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the

More information

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral. Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

More information

Math& 152 Section Integration by Parts

Math& 152 Section Integration by Parts Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

More information

Integration Techniques

Integration Techniques Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u

More information

Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1

Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1 Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show

More information

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =. Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos( - 1 2 ) = rcsin( 1 2 ) = rcsin( - 1 2 ) = Cn you do similr problems? Review of Bsic Concepts

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

More information

Main topics for the First Midterm

Main topics for the First Midterm Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

More information

Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8

Math 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8 Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite

More information

Math 116 Final Exam April 26, 2013

Math 116 Final Exam April 26, 2013 Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems

More information

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ). AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

More information

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space. Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

More information

Stuff You Need to Know From Calculus

Stuff You Need to Know From Calculus Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate

along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate L8 VECTOR EQUATIONS OF LINES HL Mth - Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne

More information

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS. THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem

More information

critical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0)

critical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0) Decoding AB Clculus Voculry solute mx/min x f(x) (sometimes do sign digrm line lso) Edpts C.N. ccelertion rte of chnge in velocity or x''(t) = v'(t) = (t) AROC Slope of secnt line, f () f () verge vlue

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

Student Session Topic: Particle Motion

Student Session Topic: Particle Motion Student Session Topic: Prticle Motion Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

1 The Riemann Integral

1 The Riemann Integral The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

More information

x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx

x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx . Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute

More information

Mathematics Extension 1

Mathematics Extension 1 04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen

More information

MAT187H1F Lec0101 Burbulla

MAT187H1F Lec0101 Burbulla Chpter 6 Lecture Notes Review nd Two New Sections Sprint 17 Net Distnce nd Totl Distnce Trvelled Suppose s is the position of prticle t time t for t [, b]. Then v dt = s (t) dt = s(b) s(). s(b) s() is

More information

First Semester Review Calculus BC

First Semester Review Calculus BC First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.

More information

Calculus AB. For a function f(x), the derivative would be f '(

Calculus AB. For a function f(x), the derivative would be f '( lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:

More information

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s). Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different

More information

Math Bootcamp 2012 Calculus Refresher

Math Bootcamp 2012 Calculus Refresher Mth Bootcmp 0 Clculus Refresher Exponents For ny rel number x, the powers of x re : x 0 =, x = x, x = x x, etc. Powers re lso clled exponents. Remrk: 0 0 is indeterminte. Frctionl exponents re lso clled

More information

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1

a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1 Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

Section Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled?

Section Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled? Section 5. - Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles

More information

First midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009

First midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009 Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

Math Calculus with Analytic Geometry II

Math Calculus with Analytic Geometry II orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

More information

Calculus II: Integrations and Series

Calculus II: Integrations and Series Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]

More information

n=0 ( 1)n /(n + 1) converges, but not n=100 1/n2, is at most 1/100.

n=0 ( 1)n /(n + 1) converges, but not n=100 1/n2, is at most 1/100. Mth 07H Topics since the second exm Note: The finl exm will cover everything from the first two topics sheets, s well. Absolute convergence nd lternting series A series n converges bsolutely if n converges.

More information

FINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+

More information

f a L Most reasonable functions are continuous, as seen in the following theorem:

f a L Most reasonable functions are continuous, as seen in the following theorem: Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f

More information

Math 113 Exam 1-Review

Math 113 Exam 1-Review Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between

More information

ROB EBY Blinn College Mathematics Department

ROB EBY Blinn College Mathematics Department ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob Eby-Fll 26 Weknowthtwhengiventhedistncefunction, wecnfindthevelocitytnypointbyfindingthederivtiveorinstntneous

More information

Math 1B, lecture 4: Error bounds for numerical methods

Math 1B, lecture 4: Error bounds for numerical methods Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

More information

Chapters 4 & 5 Integrals & Applications

Chapters 4 & 5 Integrals & Applications Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

Z b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...

Z b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but... Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.

More information

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1

More information

B Veitch. Calculus I Study Guide

B Veitch. Calculus I Study Guide Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some

More information

Precalculus Spring 2017

Precalculus Spring 2017 Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify

More information

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams.

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams. 22 TYPE PROBLEMS The AP clculus exms contin fresh crefully thought out often clever questions. This is especilly true for the free-response questions. The topics nd style of the questions re similr from

More information

MATH SS124 Sec 39 Concepts summary with examples

MATH SS124 Sec 39 Concepts summary with examples This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples

More information

Math 3B Final Review

Math 3B Final Review Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:45-10:45m SH 1607 Mth Lb Hours: TR 1-2pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems

More information

Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..

More information

Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus

Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = -x + 8x )Use

More information

The Regulated and Riemann Integrals

The Regulated and Riemann Integrals Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

1 The fundamental theorems of calculus.

1 The fundamental theorems of calculus. The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection

More information

Math 231E, Lecture 33. Parametric Calculus

Math 231E, Lecture 33. Parametric Calculus Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider

More information

Math 360: A primitive integral and elementary functions

Math 360: A primitive integral and elementary functions Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

More information

38 Riemann sums and existence of the definite integral.

38 Riemann sums and existence of the definite integral. 38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

More information

AB Calculus Path to a Five Problems

AB Calculus Path to a Five Problems AB Clculus Pth to Five Problems # Topic Completed Definition of Limit One-Sided Limits 3 Horizontl Asymptotes & Limits t Infinity 4 Verticl Asymptotes & Infinite Limits 5 The Weird Limits 6 Continuity

More information

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll

More information

Section 14.3 Arc Length and Curvature

Section 14.3 Arc Length and Curvature Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

More information

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

More information

We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.

We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b. Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn

More information

III. AB Review. The material in this section is a review of AB concepts Illegal to post on Internet

III. AB Review. The material in this section is a review of AB concepts Illegal to post on Internet III. AB Review The mteril in this section is review of AB concepts. www.mstermthmentor.com - 181 - Illegl to post on Internet R1: Bsic Differentition The derivtive of function is formul for the slope of

More information

5.7 Improper Integrals

5.7 Improper Integrals 458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

More information